In this note, our results from
[ Comm. Pure Appl. Math.63 (2010), 747--778]
on infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules
are discussed, presented in a different context, and improved by using recent observations by Morimoto and Yang.
In particular, similarities between the homogeneous Boltzmann equation
and the fractional heat equation are emphasized.
Moreover, we show that a certain conjecture by
Bobylev and Cercignani
on regularity of self-similar solutions to
the homogeneous Boltzmann equation for Maxwellian-type molecules
has a positive answer.